Geometric growth is a time-based process that increases quantity. It happens when a quantity’s instantaneous rate of change (or derivative) with respect to time is proportionate to the quantity itself. A quantity exhibiting exponential growth is described as an exponential function of time, in which the variable denoting time is the exponent (in contrast to other types of growth, such as quadratic growth).
If the proportionality constant is negative, the amount declines with time and is referred to as exponential decay instead. It’s also known as geometric growth or geometric decay when the function values exhibit a geometric progression in a discrete domain of definition with equal intervals.
The formula for exponential growth of a variable x at a certain growth rate r over time t in discrete intervals (that is, at integer periods 0, 1, 2, 3,…) is
xt = x0(1+r)t
where x0 is the value of x at time 0. A bacterial colony’s growth is commonly used to demonstrate it. One bacterium divides into two, each of which divides into four, eight, sixteen, thirty-two, and so on. Because the rate of increase is proportionate to the increasing amount of bacteria, it keeps rising. Real-life actions or phenomena, such as the spread of virus infection, the rise of debt owing to compound interest, and the spread of viral films, show this type of growth. In practise, initial exponential growth rarely lasts indefinitely, instead slowing down due to upper limitations imposed by external forces and transitioning to logistic growth.
The terms “exponential growth” and “rapid growth” are frequently misunderstood. Indeed, something that expands exponentially may first seem to grow slowly.
Examples
Biology
The number of microorganisms in a culture will multiply exponentially until a vital ingredient is depleted, at which stage there will be no more nutrition available for additional organisms to grow. Normally, the first organism splits into two daughter organisms, which then split into four, eight, and so forth. Because exponential growth suggests a steady rate of expansion, exponentially expanding cells are frequently assumed to be in a stable state. Cells, on the other hand, can grow at a consistent exponential pace while changing their metabolism and gene expression.
If no artificial vaccination is available, a virus (such as COVID-19 or smallpox) will spread exponentially at first. Each infected individual has the ability to infect a large number of additional persons.
Physics
Avalanche breakdown in a dielectric material. When a free electron collides with atoms or molecules in the dielectric medium, it is sufficiently accelerated by an externally supplied electrical field to free up more electrons. These secondary electrons are likewise accelerated, resulting in more free electrons. The consequent exponential accumulation of electrons and ions might cause the material’s dielectric breakdown to occur quickly.
Chain reaction in nuclear reactors (the concept behind nuclear reactors and nuclear weapons). Each fissioning uranium nucleus releases numerous neutrons, which can be absorbed by nearby uranium atoms, leading them to fission as well. The production rate of neutrons and induced uranium fissions rises exponentially if the chance of neutron absorption exceeds the probability of neutron escape (a function of the shape and mass of the uranium) “At any stage in the chain reaction, due to the exponential rate of increase, In the last 4.6 generations, 99 percent of the energy will have been released. The first 53 generations might be thought of as a latency phase building up to the real explosion, which takes just 3–4 generations.”
Economics
Economic growth is measured in percentages, meaning that it is exponential.
Finance
At a fixed interest rate, compound interest produces capital growth that is exponential.
This sort of growth is also seen in pyramid schemes and Ponzi schemes, resulting in significant gains for a few early investors and losses for a large number of investors.
Internet Phenomena
Internet memes and videos, for example, may spread exponentially, and are sometimes referred to as “going viral” as an analogue to the spread of infections.
With media like social networks, one person can send the same content to a large number of people at once, who then spread it to even more people, and so on, resulting in rapid spread. For example, on July 15, 2012, the video Gangnam Style was released to YouTube, garnering hundreds of thousands of views on the first day, millions on the twentieth day, and hundreds of millions in less than two months.
Conclusion
Because all populations of organisms have the potential to expand geometrically, the concept of geometric growth is of special relevance in population biology. All living creatures have a high biotic potential, or maximal rate of reproduction, which means that all species potentially have the ability to replicate themselves many, many times during their existence. In fact, due to reproductive failure, limited room and food, infections, predation, and other misfortunes, only a small proportion of most species’ offspring survive. A few species, such as the lemming, have exponential population growth cycles that culminate in extreme overpopulation. Following this, there occurs a catastrophic dieback, during which the population is drastically reduced, preparing it for the next cycle of growth and dieback. Interacting species’ population numbers will vary in a similar way. Environmental resistance, on the other hand, keeps populations steady in most cases, unless there is an environmental disturbance.